Assets,Portfolios,andArbitrage€¦ · tion1.2in order to achieve absence of arbitrage. For...

Chapter 1Assets, Portfolios, and Arbitrage

In this chapter, the concepts of portfolio, arbitrage, market completeness,pricing and hedging, are introduced in a simplified single-step financial modelwith only two time instants t = 0 and t = 1. A binary asset price model isconsidered as an example in Section 1.7.

1.1 Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . 191.2 Portfolio Allocation and Short Selling . . . . . . . . . . . 201.3 Arbitrage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4 Risk-Neutral Probability Measures . . . . . . . . . . . . . . 271.5 Hedging Contingent Claims . . . . . . . . . . . . . . . . . . . . . 301.6 Market Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . 331.7 Example: Binary Market . . . . . . . . . . . . . . . . . . . . . . . 33Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.1 Definitions and Notation

We will use the following notation. An element x of Rd+1 is a vector

x = (x(0),x(1), . . . ,x(d))

made of d+ 1 components. The scalar product x • y of two vectors x, y ∈ Rd+1

is defined by

x • y := x(0)y(0) + x(1)y(1) + · · ·+ x(d)y(d).

The vectorS0 =

(S(0)0 ,S(1)

0 , . . . ,S(d)0)

denotes the prices at time t = 0 of d+ 1 assets. Namely, S(i)0 > 0 is the price

at time t = 0 of asset no i = 0, 1, . . . , d.

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This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html

https://www.ntu.edu.sg/home/nprivault/indext.html

N. Privault

The asset values S(i)1 > 0 of assets No i = 0, 1, . . . , d at time t = 1 are

represented by the vector

S1 =(S(0)1 ,S(1)

1 , . . . ,S(d)1),

whose components(S(1)1 , . . . ,S(d)

1)are random variables defined on a proba-

bility space (Ω,F , P).

In addition we will assume that asset no 0 is a riskless asset (of savingsaccount type) that yields an interest rate r > 0, i.e. we have

S(0)1 = (1 + r)S

(0)0 .

1.2 Portfolio Allocation and Short Selling

A portfolio based on the assets 0, 1, . . . , d is viewed as a vector

ξ =(ξ(0), ξ(1), . . . , ξ(d)

)∈ Rd+1,

in which ξ(i) represents the (possibly fractional) quantity of asset no i ownedby an investor, i = 0, 1, . . . , d. The price of such a portfolio, or the cost ofthe corresponding investment, is given by

ξ • S0 =d∑i=0

ξ(i)S(i)0 = ξ(0)S

(0)0 + ξ(1)S

(1)0 + · · ·+ ξ(d)S

(d)0

at time t = 0. At time t = 1 ,the value of the portfolio has evolved into

ξ • S1 =d∑i=0

ξ(i)S(i)1 = ξ(0)S

(0)1 + ξ(1)S

(1)1 + · · ·+ ξ(d)S

(d)1 .

There are various ways to construct a portfolio allocation(ξ(i))i=0,1,...,d.

i) If ξ(0) > 0, the investor puts the amount ξ(0)S(0)0 > 0 on a savings ac-

count with interest rate r.

ii) If ξ(0) < 0, the investor borrows the amount −ξ(0)S(0)0 > 0 with the

same interest rate r.

iii) For i = 1, 2, . . . , d, if ξ(i) > 0 then the investor purchases a (possiblyfractional) quantity ξ(i) > 0 of the asset no i.

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Assets, Portfolios and Arbitrage

iv) If ξ(i) < 0, the investor borrows a quantity −ξ(i) > 0 of asset i and sellsit to obtain the amount −ξ(i)S(i)

0 > 0.

In the latter case one says that the investor short sells a quantity −ξ(i) > 0of the asset no i, which lowers the cost of the portfolio.

Definition 1.1. The short selling ratio, or percentage of daily turnover ac-tivity related to short selling, is defined as as the ratio of the number of dailyshort sold shares divided by daily volume.

Profits are usually made by first buying at a low price and then selling at ahigh price. Short sellers apply the same rule but in the reverse time order: firstsell high, and then buy low if possible, by applying the following procedure.

1. Borrow the asset no i.

2. At time t = 0, sell the asset no i on the market at the price S(i)0 and

invest the amount S(i)0 at the interest rate r > 0.

3. Buy back the asset no i at time t = 1 at the price S(i)1 , with hopefully

S(i)1 < (1 + r)S

(i)0 .

4. Return the asset to its owner, with possibly a (small) fee p > 0.∗

At the end of the operation the profit made on share no i equals

(1 + r)S(i)0 − S

(i)1 − p > 0,

which is positive provided that S(i)1 < (1 + r)S

(i)0 and p > 0 is sufficiently

small.

1.3 Arbitrage

Arbitrage can be described as:

“the purchase of currencies, securities, or commodities in one market forimmediate resale in others in order to profit from unequal prices”.†

In other words, an arbitrage opportunity is the possibility to make a strictlypositive amount of money starting from zero, or even from a negative amount.In a sense, the existence of an arbitrage opportunity can be seen as a way to“beat” the market.

∗ The cost p of short selling will not be taken into account in later calculations.† https://www.collinsdictionary.com/dictionary/english/arbitrage

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https://www.collinsdictionary.com/dictionary/english/arbitrage

https://www.collinsdictionary.com/dictionary/english/arbitrage


N. Privault

For example, triangular arbitrage is a way to realize arbitrage opportuni-ties based on discrepancies in the cross exchange rates of foreign currencies,as seen in Figure 1.1.∗

(a) (Wikipedia).

.

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(b) Cross exchange rates.

Fig. 1.1: Examples of triangular arbitrage.

As an attempt to realize triangular arbitrage based on the data of Figure 1.1b,one could:1. Change US$1.00 into €0.89347,2. Change €0.89347 into £0.89347× 0.86167 = £0.769876295,3. Change back £0.769876295 into US$0.769876295×1.2981 = US$0.999376418,which would actually result into a small loss. Alternatively, one could:1. Change US$1.00 into £0.76988,2. Change £0.76988 into €1.16054× 0.76988 =€0.893476535,3. Change back €0.893476535 into US$0.893476535×1.11923 = US$1.000005742,which would result into a small gain, assuming the absence of transactioncosts.

Next, we state a mathematical formulation of the concept of arbitrage.Definition 1.2. A portfolio allocation ξ ∈ Rd+1 constitutes an arbitrageopportunity if the three following conditions are satisfied:i) ξ • S0 6 0 at time t = 0, [start from a zero-cost portfolio or in debt]

ii) ξ • S1 > 0 at time t = 1, [finish with a nonnegative amount]

iii) P(ξ • S1 > 0

)> 0 at time t = 1. [profit made with nonzero probability]

Note that there exist multiple ways to break the assumptions of Defini-tion 1.2 in order to achieve absence of arbitrage. For example, under ab-sence of arbitrage, satisfying Condition (i) means that either ξ • S1 cannot∗ https://en.wikipedia.org/wiki/Triangular_arbitrage

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https://en.wikipedia.org/wiki/Triangular_arbitrage





be almost surely∗ nonnegative (i.e., potential losses cannot be avoided), orP(ξ • S1 > 0

)= 0, (i.e., no strictly positive profit can be made).

Realizing arbitrage

In the example below we realize arbitrage by buying and holding an asset.

1. Borrow the amount −ξ(0)S(0)0 > 0 on the riskless asset no 0.

2. Use the amount−ξ(0)S(0)0 > 0 to purchase a quantity ξ(i) = −ξ(0)S(0)

0 /S(i)0 ,

of the risky asset no i > 1 at time t = 0 and price S(i)0 so that the initial

portfolio cost isξ(0)S

(0)0 + ξ(i)S

(i)0 = 0.

3. At time t = 1, sell the risky asset no i at the price S(i)1 , with hopefully

S(i)1 > (1 + r)S

(i)0 .

4. Refund the amount −(1 + r)ξ(0)S(0)0 > 0 with interest rate r > 0.

At the end of the operation the profit made is

ξ(i)S(i)1 −

(− (1 + r)ξ(0)S

(0)0)= ξ(i)S

(i)1 + (1 + r)ξ(0)S

(0)0

= −ξ(0)S(0)0

S(i)0

S(i)1 + (1 + r)ξ(0)S

(0)0

= −ξ(0)S(0)0

S(i)0

(S(i)1 − (1 + r)S

(i)0)

= ξ(i)(S(i)1 − (1 + r)S

(i)0)

> 0,

or S(i)1 − (1 + r)S

(i)0 per unit of stock invested, which is positive provided

that S(i)1 > S

(i)0 and r is sufficiently small.

Arbitrage opportunities can be similarly realized using the short selling pro-cedure described in Section 1.2.

∗ “Almost surely”, or “a.s.”, means “with probability one”.

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N. Privault

City Currency US$Tokyo 38,800 yen $346Hong Kong HK$2,956.67 $381Seoul 378,533 won $400Taipei NT$12,980 $404New York $433Sydney A$633.28 $483Frankfurt €399 $513Paris €399 $513Rome €399 $513Brussels €399.66 $514London £279.99 $527Manila 29,500 pesos $563Jakarta 5,754,1676 rupiah $627

Fig. 1.2: Arbitrage: Retail prices around the world for the Xbox 360 in 2006.

There are many real-life examples of situations where arbitrage opportunitiescan occur, such as:

- assets with different returns (finance),

- servers with different speeds (queueing, networking, computing),

- highway lanes with different speeds (driving).

In the latter two examples, the absence of arbitrage is consequence of thefact that switching to a faster lane or server may result into congestion, thusannihilating the potential benefit of the shift.

Table 1.1: Absence of arbitrage - the Mark Six “Investment Table”.

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In the table of Figure 1.1 the absence of arbitrage opportunities is material-ized by the fact that the price of each combination is found to be proportionalto its probability, thus making the game fair and disallowing any opportunityor arbitrage that would result of betting on a more profitable combination.

In the sequel, we will work under the assumption that arbitrage opportunitiesdo not occur and we will rely on this hypothesis for the pricing of financialinstruments.

Example: share rights

Let us give a market example of pricing by absence of arbitrage.

From March 24 to 31, 2009, HSBC issued rights to buy shares at the price of$28. This right behaves similarly to an option in the sense that it gives theright (with no obligation) to buy the stock at the discount price K = $28.On March 24, the HSBC stock price closed at $41.70.

The question is: how to value the price $R of the right to buy one share? Thisquestion can be answered by looking for arbitrage opportunities. Indeed, theunderlying stock can be purchased in two different ways:1. Buy the stock directly on the market at the price of $41.70. Cost: $41.70,

or:

2. First, purchase the right at price $R, and then the stock at price $28.Total cost: $R+$28.

a) In case$R+ $28 < $41.70, (1.1)

arbitrage would be possible for an investor who owns no stock and norights, by

i) Buying the right at a price $R, and thenii) Buying the stock at price $28, andiii) Reselling the stock at the market price of $41.70.

The profit made by this investor would equal

$41.70− ($R+ $28) > 0.

b) On the other hand, in case

$R+ $28 > $41.70, (1.2)

arbitrage would be possible for an investor who owns the rights, by:" 25



N. Privault

i) Buying the stock on the market at $41.70,ii) Selling the right by contract at the price $R, and theniii) Selling the stock at $28 to that other investor.

In this case, the profit made would equal

$R+ $28− $41.70 > 0.

In the absence of arbitrage opportunities, the combination of (1.1) and(1.2) implies that $R should satisfy

$R+ $28− $41.70 = 0,

i.e. the arbitrage price of the right is given by the equation

$R = $41.70− $28 = $13.70. (1.3)

Interestingly, the market price of the right was $13.20 at the close of the ses-sion on March 24. The difference of $0.50 can be explained by the presenceof various market factors such as transaction costs, the time value of money,or simply by the fact that asset prices are constantly fluctuating over time.It may also represent a small arbitrage opportunity, which cannot be at allexcluded. Nevertheless, the absence of arbitrage argument (1.3) prices theright at $13.70, which is quite close to its market value. Thus the absence ofarbitrage hypothesis appears as an accurate tool for pricing.

Exercise: A company issues share rights, so that ten rights allow one to pur-chase three shares at the price of €6.35. Knowing that the stock is currentlyvalued at €8, estimate the price of the right by absence of arbitrage.

Answer: Letting R denote the price of one right, it will require 10R/3 topurchase one stock at €6.35, hence absence of arbitrage tells us that

103 R+ 6.35 = 8,

from which it follows that

R =310 (8− 6.35) = €0.495.

Note that the actual share right was quoted at €0.465 according to marketdata. See also Exercise 17.7 for the pricing of convertible bonds.

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1.4 Risk-Neutral Probability Measures

In order to use absence of arbitrage in the general context of pricing financialderivatives, we will need the notion of risk-neutral probability measure.

The next definition says that under a risk-neutral probability measure,the risky assets no 1, 2, . . . , d have same average rate of return as the risklessasset no 0.Definition 1.3. A probability measure P∗ on Ω is called a risk-neutral mea-sure if

IE∗[S(i)1]= (1 + r)S

(i)0 , i = 1, 2, . . . , d. (1.4)

Here, IE∗ denotes the expectation under the probability measure P∗. Notethat for i = 0, the condition IE∗

[S(0)1]= (1 + r)S

(0)0 is always satisfied by

definition.

In other words, P∗ is called “risk neutral” because taking risks under P∗

by buying a stock S(i)1 has a neutral effect: on average the expected yield of

the risky asset equals the risk-free interest rate obtained by investing on thesavings account with interest rate r.

On the other hand, under a “risk premium” probability measure P#, theexpected return of the risky asset S(i)

1 would be higher than r, i.e. we wouldhave

IE# [S(i)1]> (1 + r)S

(i)0 , i = 1, 2, . . . , d,

whereas under a “negative premium” measure P[, the expected return of therisky asset S(i)

1 would be lower than r, i.e. we would have

IE[[S(i)1]< (1 + r)S

(i)0 , i = 1, 2, . . . , d.

In the sequel we will only consider probability measures P∗ that are equivalentto P, in the sense that they share the same events of zero probability.Definition 1.4. A probability measure P∗ on (Ω,F) is said to be equivalentto another probability measure P when

P∗(A) = 0 if and only if P(A) = 0, for all A ∈ F . (1.5)

The following Theorem 1.5 can be used to check for the existence of arbitrageopportunities, and is known as the first fundamental theorem of asset pricing.Theorem 1.5. A market is without arbitrage opportunity if and only if itadmits at least one equivalent risk-neutral probability measure P∗.Proof. (i) Sufficiency. Assume that there exists a risk-neutral probabilitymeasure P∗ equivalent to P. Since P∗ is risk neutral we have" 27



N. Privault

ξ • S0 =d∑i=0

ξ(i)S(i)0 =

11 + r

d∑i=0

ξ(i) IE∗[S(i)1]=

11 + r

IE∗[ξ • S1

]> 0, (1.6)

by Definition 1.2-(ii). We proceed by contradiction, and suppose that themarket admits an arbitrage opportunity. In this case, Definition 1.2-(iii)implies P

(ξ • S1 > 0

)> 0, hence P∗

(ξ • S1 > 0

)> 0 because P is equivalent

to P∗. Since by Relation (23.13) we have

0 < P∗(ξ • S1 > 0

)= P∗

( ⋃n>1

ξ • S1 > 1/n

)= lim

n→∞P∗(ξ • S1 > 1/n

)= lim

ε0P∗(ξ • S1 > ε

),

there exists ε > 0 such that P∗(ξ • S1 > ε

)> 0, hence

IE∗[ξ • S1

]> IE∗

[ξ • S11ξ•S1>ε

]> ε IE∗

[1ξ•S1>ε

]= εP∗

(ξ • S1 > ε

)> 0,

and by (1.6) we conclude that

ξ • S0 =1

1 + rIE∗[ξ • S1

]> 0,

which results into a contradiction by Definition 1.2-(i). We conclude that themarket is without arbitrage opportunities.

(ii) The proof of necessity relies on the theorem of separation of convex setsby hyperplanes Proposition 1.6 below, see Theorem 1.6 in Föllmer and Schied(2004). It can be briefly sketched as follows. Given two financial assets withnet discounted gains X,Y and a portfolio made of one unit of X and c unit(s)of Y , the absence of arbitrage opportunity property of Definition 1.2 can bereformulated by saying that for any portfolio choice determined by c ∈ R, wehave

X + cY > 0 =⇒ X + cY = 0, P− a.s., (1.7)

i.e. a risk-free portfolio with no loss cannot entail a stricly positive gain. Inother words, if one wishes to make a strictly positive gain on the market, one

28 "




has to accept the possibility of a loss.

To show that this implies the existence of a risk-neutral probability measureP∗ under which the risky investments have zero discounted return, i.e.

IEP∗ [X ] = IEP∗ [Y ] = 0, (1.8)

the convex separation Proposition 1.6 below is applied to the convex subset

C =(IEQ[X ], IEQ[Y ]) : Q ∈ P

⊂ R2

of R2, where P is the family of probability measures Q on Ω equivalent toP. If (1.8) does not hold under any P∗ ∈ P then (0, 0) /∈ C and the convexseparation Proposition 1.6 below applied to the convex sets C and (0, 0)shows the existence of c ∈ R such that

IEQ[X ] + c IEQ[Y ] > 0 for all Q ∈ P, (1.9)

andIEP∗ [X ] + c IEP∗ [Y ] > 0 for some P∗ ∈ P. (1.10)

Condition (1.9) shows that X + cY > 0 almost surely∗ while Condition (1.10)implies P∗(X + cY > 0) > 0, which contradicts absence of arbitrage byDefinition 1.2-(iii). If the direction of the inequalities in (1.9) and (1.10) arereversed, we can reach the same conclusion by replacing the allocation (1, c)with (−1,−c).

Next is a version of the separation theorem for convex sets, which relies onthe more general Theorem 1.7 below.

Proposition 1.6. Let C be a convex set in R2 such that (0, 0) /∈ C. Thenthere exists c ∈ R such that e.g.

x+ cy > 0,

for all (x, y) ∈ C, andx∗ + cy∗ > 0,

for some (x∗, y∗) ∈ C, up to a change of direction in both inequalities > and>.

Proof. Theorem 1.7 below applied to C1 := (0, 0) and C2 := C shows that forsome a, c ∈ R we have e.g.

0 + 0× c = 0 6 a 6 x+ cy

for all (x, y) ∈ C.∗ “Almost surely”, or “a.s.”, means “with probability one”.

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N. Privault

x

y

(0, 0)

C

On the other hand, if x+ cy = 0 for all (x, y) ∈ C then the convex set C iscontained in a line crossing (0, 0), for which there clearly exist c ∈ R suchthat x+ cy > 0 for all (x, y) ∈ C and x∗ + cy∗ > 0 for some (x∗, y∗) ∈ C,because (0, 0) /∈ C.

The proof of Proposition 1.6 relies on the following result, see e.g. Theo-rem 4.14 in Hiriart-Urruty and Lemaréchal (2001).Theorem 1.7. Let C1 and C2 be two disjoint convex sets in R2. Then thereexists a, c ∈ R such that

x(1) + cy(1) 6 a and a 6 x(2) + cy(2),

for all (x(1), y(1)) ∈ C1 and (x(2), y(2)) ∈ C2 (up to exchange of C1 and C2).

C1

C2

Fig. 1.3: Separation of convex sets by the linear equation x+ cy = a.

1.5 Hedging Contingent Claims

In this section we consider the notion of contingent claim. “Contingent” isan adjective that means:30 "




1. Subject to chance.

2. Occurring or existing only if (certain circumstances) are the case; depen-dent on.

More generally, we will work according to the following broad definition.

Definition 1.8. A contingent claim is any nonnegative random variable C :Ω −→ [0,∞).

In practice, the random variable C will represent the payoff of an (option)contract at time t = 1.

Referring to Definition 0.2, the European call option with maturity t = 1on the asset no i is a contingent claim whose payoff C is given by

C =(S(i)1 −K

)+ :=

S(i)1 −K if S(i)

1 > K,

0 if S(i)1 < K,

where K is called the strike price. The claim payoff C is called “contingent”because its value may depend on various market conditions, such as S(i)

1 > K.A contingent claim is also called a financial “derivative” for the same reason.

Similarly, referring to Definition 0.1, the European put option with matu-rity t = 1 on the asset no i is a contingent claim with payoff

C =(K − S(i)

1)+ :=

K − S(i)

1 if S(i)1 6 K,

0 if S(i)1 > K,

Definition 1.9. A contingent claim with payoff C is said to be attainable ifthere exists a portfolio allocation ξ = (ξ(0), ξ(1), . . . , ξ(d)) such that

C = ξ • S1 =d∑i=0

ξ(i)S(i)1 = ξ(0)S

(0)1 + ξ(1)S

(1)1 + · · ·+ ξ(d)S

(d)1 ,

with P-probability one.

When a contingent claim with payoff C is attainable, a trader will be ableto:

1. at time t = 0, build a portfolio allocation ξ = (ξ(0), ξ(1), . . . , ξ(d)) ∈Rd+1,

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N. Privault

2. invest the amount

ξ • S0 =d∑i=0

ξ(i)S(i)0

in this portfolio at time t = 0,

3. at time t = 1, obtain the equality

C =d∑i=0

ξ(i)S(i)1

and pay the claim amount C using the portfolio value

ξ • S1 =d∑i=0

ξ(i)S(i)1 = ξ(0)S

(0)1 + ξ(1)S

(1)1 + · · ·+ ξ(d)S

(d)1 .

We note that in order to attain the claim payoff C, an initial investmentξ • S0 is needed at time t = 0. This amount, to be paid by the buyer to theissuer of the option (the option writer), is also called the arbitrage price, oroption premium, of the contingent claim, and is denoted by

π0(C) := ξ • S0 =d∑i=0

ξ(i)S(i)0 . (1.11)

The action of allocating a portfolio allocation ξ such that

C = ξ • S1 =d∑i=0

ξ(i)S(i)1 (1.12)

is called hedging, or replication, of the contingent claim with payoff C.

Definition 1.10. In case the portfolio value ξ • S1 at time t = 1 exceeds theamount of the claim, i.e. when

ξ • S1 > C,

we say that the portfolio allocation ξ = (ξ(0), ξ(1), . . . , ξ(d)) is super-hedging.

In this document we only focus on hedging, i.e. on replication of the contin-gent claim with payoff C, and we will not consider super-hedging.

As a simplified illustration of the principle of hedging, one may an buyoil-related asset in order to hedge oneself against a potential price rise ofgasoline. In this case, any increase in the price of gasoline that would resultin a higher value of the financial derivative C would be correlated to an

32 "




increase in the underlying asset value, so that the equality (1.12) would bemaintained.

1.6 Market Completeness

Market completeness is a strong property saying that any contingent claimcan be perfectly hedged.

Definition 1.11. A market model is said to be complete if every contingentclaim is attainable.

The next result is the second fundamental theorem of asset pricing.

Theorem 1.12. A market model without arbitrage opportunities is completeif and only if it admits only one equivalent risk-neutral probability measureP∗.

Proof. See the proof of Theorem 1.40 in Föllmer and Schied (2004).

Theorem 1.12 will give us a concrete way to verify market completeness bysearching for a unique solution P∗ to Equation (1.4).

1.7 Example: Binary Market

In this section we work out a simple example that allows us to apply Theo-rem 1.5 and Theorem 1.12. We take d = 1, i.e. the portfolio is made of

- a riskless asset with interest rate r and priced (1+ r)S(0)0 at time t = 1,

- and a risky asset priced S(1)1 at time t = 1.

We use the probability space

Ω = ω−,ω+,

which is the simplest possible nontrivial choice of probability space, made ofonly two possible outcomes with

P(ω−) > 0 and P(ω+) > 0,

in order for the setting to be nontrivial. In other words the behavior of themarket is subject to only two possible outcomes, for example, one is expect-ing an important binary decision of “yes/no” type, which can lead to twodistinct scenarios called ω− and ω+.

In this context, the asset price S(1)1 is given by a random variable

S(1)1 : Ω −→ R

" 33



N. Privault

whose value depends on whether the scenario ω−, resp. ω+, occurs.

Precisely, we set

S(1)1 (ω−) = a, and S

(1)1 (ω+) = b,

i.e. the value of S(1)1 becomes equal a under the scenario ω−, and equal to b

under the scenario ω+, where 0 < a < b. ∗

Arbitrage

The first natural question is:- Arbitrage: Does the market allow for arbitrage opportunities?

We will answer this question using Theorem 1.5, by searching for a risk-neutral probability measure P∗ satisfying the relation

IE∗[S(1)1]= (1 + r)S

(1)0 , (1.13)

where r > 0 denotes the risk-free interest rate, cf. Definition 1.3.

In this simple framework, any measure P∗ on Ω = ω−,ω+ is characterizedby the data of two numbers P∗(ω−) ∈ [0, 1] and P∗(ω+) ∈ [0, 1], suchthat

P∗(Ω) = P∗(ω−) + P∗(ω+) = 1. (1.14)

Here, saying that P∗ is equivalent to P simply means that

P∗(ω−) > 0 and P∗(ω+) > 0.

Although we should solve (1.13) for P∗, at this stage it is not yet clear howP∗ is involved in the equation. In order to make (1.13) more explicit we writethe expected value as

IE∗[S(1)1]= aP∗

(S(1)1 = a

)+ bP∗

(S(1)1 = b

),

hence Condition (1.13) for the existence of a risk-neutral probability measureP∗ reads

aP∗(S(1)1 = a

)+ bP∗

(S(1)1 = b

)= (1 + r)S

(1)0 .

Using the Condition (1.14) we obtain the system of two equationsaP∗(ω−) + bP∗(ω+) = (1 + r)S(1)0

P∗(ω−) + P∗(ω+) = 1,(1.15)

∗ The case a = b leads to a trivial, constant market.

34 "




with unique risk-neutral solution

p∗ := P∗(ω+) = P∗

(S(1)1 = b

)=

(1 + r)S(1)0 − a

b− a

q∗ := P∗(ω−) = P∗(S(1)1 = a

)=b− (1 + r)S

(1)0

b− a.

(1.16)

In order for a solution P∗ to exist as a probability measure, the numbersP∗(ω−) and P∗(ω+) must be nonnegative. In addition, for P∗ to beequivalent to P they should be strictly positive from (1.5).

We deduce that if a, b and r satisfy the condition

a < (1 + r)S(1)0 < b, (1.17)

then there exists a risk-neutral equivalent probability measure P∗ which isunique, hence by Theorems 1.5 and 1.12 the market is without arbitrage andcomplete.

Remark 1.13. i) If a = (1+ r)S(1)0 , resp. b = (1+ r)S

(1)0 , then P∗(ω+) =

0, resp. P∗(ω−) = 0, and P∗ is not equivalent to P.

Therefore, we check from (1.16) that the conditions

a < b 6 (1 + r)S(1)0 or (1 + r)S

(1)0 6 a < b, (1.18)

do not imply existence of an equivalent risk-neutral probability measureand absence of arbitrage opportunities in general.

ii) If a = b = (1 + r)S(1)0 then (1.4) admits an infinity of solutions, hence

the market is without arbitrage but it is not complete. More precisely, inthis case both the riskless and risky assets yield a deterministic returnrate r and the portfolio value becomes

ξ • S1 = (1 + r)ξ • S0,

at time t = 1, hence the terminal value ξ • S1 is deterministic and thissingle value can not always match the value of a contingent claim with(random) payoff C, that could be allowed to take two distinct valuesC(ω−) and C(ω+). Therefore, market completeness does not hold whena = b = (1 + r)S

(1)0 .

Let us give a financial interpretation of Condition (1.18).

" 35



N. Privault

1. If (1 + r)S(1)0 6 a < b, let ξ(1) := 1 and choose ξ(0) such that

ξ(0)S(0)0 + ξ(1)S

(1)0 = 0

according to Definition 1.2-(i), i.e.

ξ(0) = −ξ(1)S(1)0 /S(0)

0 < 0.

(1 + r)S(1)0S

(1)0

b

a

In particular, Condition (i) of Definition 1.2 is satisfied, and the investorborrows the amount −ξ(0)S(0)

0 > 0 on the riskless asset and uses it tobuy one unit ξ(1) = 1 of the risky asset. At time t = 1 she sells therisky asset S(1)

1 at a price at least equal to a and refunds the amount−(1 + r)ξ(0)S

(0)0 > 0 she borrowed, with interest. Her profit is

ξ • S1 = (1 + r)ξ(0)S(0)0 + ξ(1)S

(1)1

> (1 + r)ξ(0)S(0)0 + ξ(1)a

= −(1 + r)ξ(1)S(1)0 + ξ(1)a

= ξ(1)(− (1 + r)S

(1)0 + a

)> 0, · ·

^

which satisfies Condition (ii) of Definition 1.2. In addition, Condition (iii)of Definition 1.2 is also satisfied because

P(ξ • S1 > 0

)= P

(S(1)1 = b

)= P(ω+) > 0.

2. If a < b 6 (1 + r)S(1)0 , let ξ(0) > 0 and choose ξ(1) such that

ξ(0)S(0)0 + ξ(1)S

(1)0 = 0,

according to Definition 1.2-(i), i.e.

ξ(1) = −ξ(0)S(0)0 /S(1)

0 < 0.

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(1 + r)S(1)0

S(1)0

b

a

This means that the investor borrows a (possibly fractional) quantity−ξ(1) > 0 of the risky asset, sells it for the amount −ξ(1)S(1)

0 , and in-vests this money on the riskless account for the amount ξ(0)S(0)

0 > 0. Asmentioned in Section 1.2, in this case one says that the investor shortsellsthe risky asset. At time t = 1 she obtains (1 + r)ξ(0)S

(0)0 > 0 from the

riskless asset and she spends at most b to buy the risky asset and returnit to its original owner. Her profit is

ξ • S1 = (1 + r)ξ(0)S(0)0 + ξ(1)S

(1)1

> (1 + r)ξ(0)S(0)0 + ξ(1)b

= −(1 + r)ξ(1)S(1)0 + ξ(1)b

= ξ(1)(− (1 + r)S

(1)0 + b

)> 0, · ·

^

since ξ(1) < 0. Note that here, a 6 S(1)1 6 b became

ξ(1)b 6 ξ(1)S(1)1 6 ξ(1)a

because ξ(1) < 0. We can check as in Part 1 above that Conditions (i)-(iii)of Definition 1.2 are satisfied.

3. Finally if a = b 6= (1+ r)S(1)0 then (1.4) admits no solution as a probability

measure P∗ hence arbitrage opportunities exist and can be constructed bythe same method as above.

Under Condition (1.17) there is absence of arbitrage and Theorem 1.5 showsthat no portfolio strategy can yield both ξ • S1 > 0 and P

(ξ • S1 > 0

)> 0

starting from ξ(0)S(0)0 + ξ(1)S

(1)0 6 0, however this is less simple to show

directly.

Market completeness

The second natural question is:

- Completeness: Is the market complete, i.e., are all contingent claimsattainable?

" 37



N. Privault

In the sequel we work under the condition

a < (1 + r)S(1)0 < b,

under which Theorems 1.5 and 1.12 show that the market is without arbi-trage and complete since the risk-neutral probability measure P∗ exists andis unique.

Let us recover this fact by elementary calculations. For any contingentclaim with payoff C we need to show that there exists a portfolio allocationξ = (ξ(0), ξ(1)) such that C = ξ • S1, i.e.

ξ(0)(1 + r)S(0)0 + ξ(1)b = C(ω+)

ξ(0)(1 + r)S(0)0 + ξ(1)a = C(ω−).

(1.19)

These equations can be solved as

ξ(0) =bC(ω−)− aC(ω+)

S(0)0 (1 + r)(b− a)

and ξ(1) =C(ω+)−C(ω−)

b− a. (1.20)

In this case we say that the portfolio allocation (ξ(0), ξ(1)) hedges the contin-gent claim with payoff C. In other words, any contingent claim is attainableand the market is indeed complete. Here, the quantity

ξ(0)S(0)0 =

bC(ω−)− aC(ω+)

(1 + r)(b− a)

represents the amount invested on the riskless asset.

Note that if C(ω+) > C(ω−) then ξ(1) > 0 and there is not short selling.This occurs in particular if C has the form C = h

(S(1)1)with x 7−→ h(x) a

non-decreasing function, since

ξ(1) =C(ω+)−C(ω−)

b− a

=h(S(1)1 (ω+)

)− h(S(1)1 (ω−)

)b− a

=h(b)− h(a)

b− a> 0,

thus there is no short selling. This applies in particular to European calloptions with strike price K, for which the function h(x) = (x −K)+ is

38 "




non-decreasing. Similarly we will find that ξ(1) 6 0, i.e. short selling alwaysoccurs when h is a non-increasing function, which is the case in particularfor European put options with payoff function h(x) = (K − x)+.

Arbitrage price

Definition 1.14. The arbitrage price π0(C) of the contingent claim withpayoff C is defined in (1.11) as the initial value at time t = 0 of the portfoliohedging C, i.e.

π0(C) = ξ • S0 =d∑i=0

ξ(i)S(i)0 , (1.21)

where (ξ(0), ξ(1)) are given by (1.20).Arbitrage prices can be used to ensure that financial derivatives are “marked”at their fair value (mark to market).∗ Note that π0(C) cannot be 0 since thiswould entail the existence of an arbitrage opportunity according to Defini-tion 1.2.

The next proposition shows that the arbitrage price π0(C) of the claimcan be computed as the expected value of its payoff C under the risk-neutralprobability measure, after discounting by the factor 1+ r in order to accountfor the time value of money.Proposition 1.15. The arbitrage price π0(C) = ξ • S0 of the contingentclaim with payoff C is given by

π0(C) =1

1 + rIE∗[C]. (1.22)

Proof. Using the expressions (1.16) of the risk-neutral probabilities P∗(ω−),P∗(ω+), and (1.20) of the portfolio allocation (ξ(0), ξ(1)), we have

π0(C) = ξ • S0

= ξ(0)S(0)0 + ξ(1)S

(1)0

=bC(ω−)− aC(ω+)

(1 + r)(b− a)+ S

(1)0C(ω+)−C(ω−)

b− a

=1

1 + r

(C(ω−)

b− S(1)0 (1 + r)

b− a+C(ω+)

(1 + r)S(1)0 − a

b− a

)

=1

1 + r

(C(ω−)P∗

(S(1)1 = a

)+C(ω+)P∗

(S(1)1 = b

))=

11 + r

IE∗[C].

∗ Not to be confused with “market making”." 39



N. Privault

In the case of a European call option with strike price K ∈ [a, b] we haveC =

(S(1)1 −K

)+ and

π0((S(1)1 −K

)+)=

11 + r

IE∗[(S(1)1 −K

)+]=

11 + r

(b−K)P∗(S(1)1 = b

)=

11 + r

(b−K)(1 + r)S

(1)0 − a

b− a.

=b−Kb− a

(S(1)0 − a

1 + r

).

In the case of a European put option we have C =(K − S(1)

1)+ and

π0((K − S(1)

1)+)

=1

1 + rIE∗[(K − S(1)

1)+]

=1

1 + r(K − a)P∗

(S(1)1 = a

)=

11 + r

(K − a)b− (1 + r)S

(1)0

b− a.

=K − ab− a

(b

1 + r− S(1)

0

).

Here,(S(1)0 −K

)+, resp. (K − S(1)0)+ is called the intrinsic value at time 0

of the call, resp. put option.

The simple setting described in this chapter raises several questions and re-marks.

Remarks

1. The fact that π0(C) can be obtained by two different methods, i.e. analgebraic method via (1.20) and (1.21) and a probabilistic method from(1.22) is not a simple coincidence. It is actually a simple example of thedeep connection that exists between probability and analysis.

In a continuous-time setting, (1.20) will be replaced with a partial differ-ential equation (PDE) and (1.22) will be computed via the Monte Carlomethod. In practice, the quantitative analysis departments of major fi-nancial institutions can be split into a “PDE team” and a “Monte Carloteam”, often trying to determine the same option prices by two different

40 "




methods.

2. What if we have three possible scenarios, i.e. Ω = ω−,ωo,ω+ and therandom asset S(1)

1 is allowed to take more than two values, e.g. S(1)1 ∈

a, b, c according to each scenario? In this case the system (1.15) wouldbe rewritten asaP∗(ω−) + bP∗(ωo) + cP∗(ω+) = (1 + r)S

(1)0

P∗(ω−) + P∗(ωo) + P∗(ω+) = 1,

and this system of two equations with three unknowns does not admit aunique solution, hence the market can be without arbitrage but it cannotbe complete, cf. Exercise 1.4.

Market completeness can be reached by adding a second risky asset, i.e.taking d = 2, in which case we will get three equations and three un-knowns. More generally, when Ω contains n > 2 market scenarios, com-pleteness of the market can be reached provided that we consider d riskyassets with d + 1 > n. This is related to the Meta-Theorem 8.3.1 ofBjörk (2004a) in which the number d of traded risky underlying assetsis linked to the number of random sources through arbitrage and marketcompleteness.

Exercises

Exercise 1.1 Consider a risky asset valued S0 = $3 at time t = 0 and takingonly two possible values S1 ∈ $1, $5 at time t = 1, and a financial claimgiven at time t = 1 by

C :=

$0 if S1 = $5

$2 if S1 = $1.

Is C the payoff of a call option or of a put option? Give the strike price ofthe option.

Exercise 1.2 Consider a risky asset valued S0 = $4 at time t = 0, and takingonly two possible values S1 ∈ $2, $5 at time t = 1. Compute the initialvalue V0 = αS0 + $β of the portfolio hedging the claim payoff

C =

$0 if S1 = $5

$6 if S1 = $2" 41



N. Privault

at time t = 1, and find the corresponding risk-neutral probability measureP∗.

Exercise 1.3

a) Consider the following market model:

(1 + r)S(1)0S

(1)0

b

a

i) Does this model allow for arbitrage? Yes | No |

ii) If this model allows for arbitrage opportunities, how can they berealized?

By shortselling | By borrowing on savings | N.A. |

b) Consider the following market model:

b

S(1)0

(1 + r)S(1)0

a




c) Consider the following market model:

(1 + r)S(1)0

S(1)0

b

a


42 "






Exercise 1.4 In a market model with two time instants t = 0 and t = 1,consider

- a riskless asset valued S(0)0 at time t = 0, and value S(0)

1 = (1 + r)S(0)0

at time t = 1.

- a risky asset with price S(1)0 at time t = 0, and three possible values at

time t = 1, with a < b < c, i.e.:

S(1)1 =

S(1)0 (1 + a),

S(1)0 (1 + b),

S(1)0 (1 + c).

a) Show that this market is without arbitrage but not complete.b) In general, is it possible to hedge (or replicate) a claim with three distinct

claim payoff values Ca,Cb,Cc in this market?

Exercise 1.5 We consider a riskless asset valued S(0)1 = S

(0)0 , at times k = 0, 1,

where the risk-free interest rate is r = 0, and a risky asset S(1) whose returnR1 := (S

(1)1 − S(1)

0 )/S(1)0 can take three values (−b, 0, b) at each time step,

with b > 0 and

p∗ := P∗(R1 = b) > 0, θ∗ := P∗(R1 = 0) > 0, q∗ := P∗(R1 = −b) > 0,

a) Determine all possible risk-neutral probability measures P∗ equivalent toP in terms of the parameter θ∗ ∈ (0, 1) from the condition IE∗[R1] = 0.

b) We assume that the variance Var∗[S(1)1 − S(1)

0

S(1)0

]= σ2 > 0 of the asset

return is known to be equal to σ2. Show that this condition determines aunique risk-neutral probability measure P∗σ under a certain condition onb and σ.

Exercise 1.6a) Consider the following binary one-step model (St)t=0,1,2 with interest rate

r = 0 and P(S1 = 2) = 1/3." 43



N. Privault

S1 = 2

S0 = 1

S1 = 1

i) Is the model without arbitrage? Yes | No |

ii) Does there exist a risk-neutral measure P∗ equivalent to P? Yes |

No |

b) Consider the following ternary one-step model with r = 0, P(S1 = 2) =1/4 and P(S1 = 0) = 1/9.

S0 = 1

S1 = 0

S1 = 1

S1 = 2

i) Does there exist a risk-neutral measure P∗ equivalent to P? Yes |

No |

ii) Is the model without arbitrage? Yes | No |

iii) Is the market complete? Yes | No |iv) Does there exist a unique risk-neutral measure P∗ equivalent to P?

Yes | No |

Exercise 1.7 Consider a one-step market model with two time instants t = 0and t = 1 and two assets:

- a riskless asset π with price π0 at time t = 0 and value π1 = π0(1 + r)at time t = 1,- a risky asset S with price S0 at time t = 0 and random value S1 at timet = 1.

We assume that S1 can take only the values S0(1 + a) and S0(1 + b), where−1 < a < r < b. The return of the risky asset is defined as

R =S1 − S0S0

.

a) What are the possible values of R?

44 "




b) Show that under the probability measure P∗ defined by

P∗(R = a) =b− rb− a

, P∗(R = b) =r− ab− a

,

the expected return IE∗[R] of S is equal to the return r of the risklessasset.

c) Does there exist arbitrage opportunities in this model? Explain why.d) Is this market model complete? Explain why.e) Consider a contingent claim with payoff C given by

C =

α if R = a,

β if R = b.

Show that the portfolio allocation (η, ξ) defined∗ by

η =α(1 + b)− β(1 + a)

π0(1 + r)(b− a)and ξ =

β − αS0(b− a)

,

hedges the contingent claim with payoff C, i.e. show that at time t = 1we have

ηπ1 + ξS1 = C.

Hint: distinguish two cases R = a and R = b.f) Compute the arbitrage price π0(C) of the contingent claim payoff C usingη, π0, ξ, and S0.

g) Compute IE∗[C] in terms of a, b, r,α,β.h) Show that the arbitrage price π0(C) of the contingent claim with payoff

C satisfiesπ0(C) =

11 + r

IE∗[C]. (1.23)

i) What is the interpretation of Relation (1.23) above?j) Let C denote the payoff at time t = 1 of a put option with strike price

K=$11 on the risky asset. Give the expression of C as a function of S1and K.

k) Letting π0 = S0 = 1, r = 5% and a = 8, b = 11, α = 2, β = 0, computethe portfolio allocation (ξ, η) hedging the contingent claim with payoff C.

l) Compute the arbitrage price π0(C) of the claim payoff C.

Exercise 1.8 Consider a stock valued S0 = $180 at the beginning of theyear. At the end of the year, its value S1 can be either $152 or $203 andthe risk-free interest rate is r = 3% per year. Given a put option with strike∗ Here, η is the (possibly fractional) quantity of asset π and ξ is the quantity held ofasset S.

" 45



N. Privault

price K on this underlying asset, find the value of K for which the price ofthe option at the beginning of the year is equal to the intrinsic option payoff.This value of K is called the break-even strike price. In other words, thebreak-even price is the value of K for which immediate exercise of the optionis equivalent to holding the option until maturity.

How would a decrease in the interest rate r affect this break-even strike price?

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